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Keywords:
boundary value problem; existence of solutions; ordinary differential equations in Hilbert space; lower and upper solution
boundary value problem; existence of solutions; ordinary differential equations in Hilbert space; lower and upper solution
Summary:
In this paper we deal with the boundary value problem in the Hilbert space. Existence of a solutions is proved by using the method of lower and upper solutions. It is not necessary to suppose that the homogeneous problem has only the trivial solution. We use some results from functional analysis, especially the fixed-point theorem in the Banach space with a cone (Theorem 4.1, [5]).
References:
[1] V. Šeda: On some non-linear boundary value problems for ordinary differential equations. Archivum Mathematicum (Brno) 25 (1989), 207-222. MR 1188065
[2] B. Rudolf: Periodic boundary value problem in Hilbert space for differential equation of second order with reflection to the argument. Mathematica Slovaca 42 no. 1 (1992), 65-84. MR 1159492
[3] M. Greguš M. Švec V. Šeda: Ordinary differential equations. Alfa, Bratislava, 1985. (In Slovak.)
[4] G. J. Šilov: Mathematical analysis. (Slovak translation), Alfa, Bratislava, 1985.
[5] M. A. Krasnosel'skij: Positive solutions of operators equations. Gosud. izd., Moskva, 1962. (In Russian.)
[6] E. Rovderová: A note on a Cone in the space $L_2(, H)$ . Diploma Thesis, 1990.
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